Download Slides for Rosen, 5th edition

Module #1 - Logic
Probability Theory
Based on Rosen, Discrete Mathematics & Its Applications. Prepared
by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Why Probability?
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In the real world, we often don’t know
whether a given proposition is true or false.
Probability theory gives us a way to reason
about propositions whose truth is
uncertain.
Useful in weighing evidence, diagnosing
problems, and analyzing situations whose
exact details are unknown.
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Random Variables
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A random variable V is a variable whose
value is unknown, or that depends on the
situation.
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e.g., the number of students in class today
Whether it will rain tonight (Boolean
variable)
Let the domain of V be dom[V]={v1,…,vn}
The proposition V=vi may be uncertain, and
be assigned a probability.
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Experiments
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A (stochastic) experiment is a process by
which a given random variable gets
assigned a specific value.
The sample space S of the experiment is
the domain of the random variable.
The outcome of the experiment is the
specific value of the random variable that is
selected.
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Events
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An event E is a set of possible outcomes.
That is, E  S = dom[V]. We say that event
E occurs when VE.
Note that VE is the (uncertain)
proposition that the actual outcome will be
one of the outcomes in the set E.
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Probability
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The probability p = Pr[E]  [0,1] of an event E is
a real number representing our degree of certainty
that E will occur.
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If Pr[E] = 1, then E is absolutely certain to occur,
 thus VE is true.
If Pr[E] = 0, then E is absolutely certain not to occur,
 thus VE is false.
If Pr[E] = ½, then we are completely uncertain about
whether E will occur; that is, VE and VE are
considered equally likely.
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Four Definitions of Probability
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Several alternative definitions of
probability are commonly encountered:
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Frequentist, Bayesian, Laplacian, Axiomatic
They have different strengths &
weaknesses.
Fortunately, they coincide and work well
with each other in most cases.
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Probability: Laplacian Definition
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First, assume that all outcomes in the sample
space are equally likely
 This term still needs to be defined.
Then, the probability of event E,
Pr[E] = |E|/|S|. Very simple!
Problems: Still needs a definition for equally
likely, and depends on existence of a finite sample
space with all equally likely outcomes.
We will use Laplace’s definition of the probability
of an event in this course.
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Probability of Complementary
Events
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Let E be an event in a sample space S.
Then, E represents the complementary event
that VE.
Pr[E ] = 1 − Pr[E]
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Probability of Unions of Events
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Let E1,E2  S = dom[V].
Then:
Pr[E1 E2] = Pr[E1] + Pr[E2] − Pr[E1E2]
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By the inclusion-exclusion principle.
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Mutually Exclusive Events
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Two events E1, E2 are called mutually
exclusive if they are disjoint: E1E2 = 
Note that two mutually exclusive events
cannot both occur in the same instance of a
given experiment.
For mutually exclusive events,
Pr[E1  E2] = Pr[E1] + Pr[E2].
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Exhaustive Sets of Events
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A set E = {E1, E2, …} of events in the
sample space S is exhaustive if
E
i
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S
An exhaustive set of events that are all
mutually exclusive with each other has the
property that
 Pr[ E ]  1
i
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Independent Events
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Two events E, F are independent if
Pr[EF] = Pr[E]·Pr[F].
Relates to product rule for number of ways
of doing two independent tasks
Example: Flip a coin, and roll a die.
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Pr[ quarter is heads  die is 1 ] =
Pr[quarter is heads] × Pr[die is 1]
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Conditional Probability
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Let E, F be events such that Pr[F]>0.
Then, the conditional probability of E given
F, written Pr[E|F], is defined as
Pr[EF]/Pr[F].
This is the probability that E would turn out
to be true, given just the information that F
is true.
If E and F are independent, Pr[E|F] = Pr[E].
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Expectation Values
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For a random variable V having a numeric
domain, its expectation value or expected
value or weighted average value or
arithmetic mean value Ex[V] is defined as
vdom[V] v·p(v)
The term “expected value” is widely used,
but misleading since the expected value
might be totally unexpected or impossible!
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Derived Random Variables
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Let S be a sample space over values of a random
variable V (representing possible outcomes).
Then, any function f over S can also be
considered to be a random variable (whose value
is derived from the value of V).
If the range R = range[f] of f is numeric, then
Ex[f] can still be defined, as
 p( s)  f ( s)
sS
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Based on Rosen, Discrete Mathematics & Its Applications. Prepared by
(c)2001-2004, Michael P. Frank and Modified By Mingwu Chen
Review: Probability
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Random Variables
Probability definition
Independent events
Conditional probability
Expectation values
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